# Tag Archives: Teaching Resources

## In the trenches with enVisionMATH

It’s been back-to-school time for everybody in our household (hence an excuse for the light posting). We started classes at the college today, and last week the 4.5-year old went back to preschool full-time and the 6.5-year old started first grade. (The 1.5-year old is rocking the local daycare.) One of the biggest changes for the kids is for our first-grader, Lucy, since she has real homework for the first time. It’s not much; the expectation is about 20 minutes a night, Monday through Thursday. Some of that homework is math, which I was very excited about — but then that excitement turned to alert caution when I learned my daughter’s class was using enVisionMATH.

I wrote this post on enVisionMATH almost three years ago, basically laughing it off the blogosphere for its happy-clappy, uncritical acceptance of unproven digital nativist frameworks and for going way over the top with smartboards. Little did I know that my own offspring would be in the middle of it just three years later. So, in an effort to process what she’s doing (for me, for her, and for anybody else who cares), this is the first of what might be many posts about the specifics of enVisionMATH, as viewed by a parent whose kid happens to be learning from that curriculum, and who also happens to be a mathematician and math teacher.

I’ll start with the worksheet Lucy brought home this evening, called “Making 8”:

I’ve never had a kid in first grade before, nor can I remember how I did this stuff in first grade, nor have I recently worked with a kid in first grade. So I’m going to share my thoughts, but realize I have no reference for what’s “normal” pedagogy for 6-year olds and what’s not.

This worksheet is really about subtraction, although it never comes out and says so. The first two exercises are attempting to build a sense about subtraction by getting kids to think about how parts fit together to form a particular quantity. enVisionMATH appears to be really big on getting kids to recognize numbers visually rather than by counting. I’ll need to blog about this in a later post, but Lucy’s had some other exercises that, for example, stress the ability to recognize this:

…as the number 6, just by looking at it and without counting the dots, almost to the point of telling kids that they shouldn’t be counting anything but rather arranging things into patterns. Again, that’s for another post.

So, back to the worksheet, kids are supposed to look at the first collection of balloons and, knowing that there are eight of them, see — and only “see” — that 8 splits into 2 plus 6, and then 4 plus 4. I did a few more of these with Lucy using coins (no balloons on hand, sadly). Biggest challenge here: Keeping Lucy from just counting the black balloons and then counting the white balloons. And the only reason this was a challenge was because, as a math person, I knew what the worksheet was getting at: recognizing quantities through visual patterns rather than counting, so the unwritten rule is for kids not to count the balloons. But other parents probably didn’t know this, and their kids just counted. I don’t think this is necessarily wrong, but it doesn’t necessarily help in the next sections either.

The next section is rather startlingly labelled “Algebra”. Remember: This is a worksheet for a first grade class. Why we are bringing up the word “algebra” at this point is anybody’s guess. I suspect this is more to make parents, school boards, and accreditors happy than it is to start getting kids to feel comfortable with the word “algebra”.  But anyway, as you can see, the two problems are just the first two problems in reverse.

Lucy had a hard time with this. First of all, she didn’t understand what “the whole” meant. This is not the first time Lucy’s struggled not with the math but with relatively esoteric vocabulary in her math lessons. Last week she had a worksheet where she was to arrange three integers “in order from greatest to least” and “from least to greatest” and we had to take a moment to figure out what all of that meant. Maybe other people’s kids don’t struggle with that, but on the other hand it’s been verified that Lucy is reading at a third or fourth grade level right now, so I wonder if it’s just her.

We had to work these out using manipulatives. We started with fingers because that’s the first thing I thought of. So, I said, if the whole is 8:

…and one part is 3:

…then what was the other part?

Lucy was able to get the answer of “5” with no problem. But… I don’t think she got it the right way. Because when we moved to the next problem and the “one part” was 1, for her, the other part was still 5! This was because when I held up one finger on my left hand this time, there were of course five fingers on the right hand. I tried holding up eight and wiggling one finger instead of putting the “one part” on one hand, but that just confused her. So, we went back to coins and built a “balloon diagram” like in the first two problems, and she got them just fine (and without counting).

I don’t think exercises 3 and 4 are bad problems necessarily, but I do think they came in here way too early. Perhaps I’m missing the context of the actual classroom interaction between Lucy and her teacher, but it would seem like a better idea to do as many exercises like 1 and 2 as possible before moving on to the “algebra”. After all, if you stick to positive integers, there are only seven ways to fill in the blanks __ + __ = 8. (And doing all seven might help kids discover the commutative property early on, which seems like a much more important thing to bring up than “algebra” in first grade.)

And then, it’s not clear to me that doing “algebra” is a better idea here than just doing straight-up subtraction.  What’s to be gained by saying “the whole is 8; one part is 3; the other part is ____” versus “What is 8 minus 3?” Again, maybe I’m out of touch, but subtraction is a fundamental skill that algebra builds upon; doing algebra before subtraction seems a little backwards to say the least. A kid who is comfortable with subtraction will be able to do these whole/part problems in a snap by using subtraction. A kid doing these “algebra” problems basically has to invent subtraction in order to do them, or else draw pictures of balloons and start counting. It feels like the curriculum is trying to be intentionally nontraditional here, just for the sake of doing things differently rather than because it works better.

Then we come to the “Journal” question, which is downright sophisticated: “The whole is 8. One part is 8. What is the other part?” Here we reach serious abstraction: You can’t draw balloons like in exercises 1 and 2, and in fact resorting to physical props is tricky.As Derek Bruff mentioned in a tweet about this earlier this evening, the use of the word “part” in conjunction with the quantity 0 is already sort of questionable. What does it even mean to say the “part” is 0? What “part”? I don’t see a “part”. The natural way of interpreting what a “part” is, is as a bunch of objects. If there are no objects present, then there really isn’t a “part”.

We had to resort to thinking not about objects but containers that hold the objects. I took two books sitting nearby. I took my eight coins and said: The whole is 8. One part is 2 — and put 2 coins on one of the books. What is the other part? — and put the remaining coins on the other book. Lucy got the right answer quickly, and she did so by looking back at exercise 1 with the balloons and noticing it was the same problem with different objects, which I thought was pretty smart. I’ll make an algebraist out of her yet! Then I repeated with one part being 1. Then I did it with one part being 6; then 7. Then I said, “The whole is 8; one part is eight.” — putting all eight coins on one book. “What’s the other part?” — showing her my empty hands and an empty book. “Zero,” she said right away.

For her, and maybe not just her, “zero” represents not a size of a part but a state of emptiness of a container — or perhaps the size of a set. It’s how much you see when nothing is there. To map the “zero” concept onto a concept of “part” that presupposes something is there just doesn’t make sense. If this sounds like the New Math, I think we’re barking up the right tree.

The “Tell how you know” was especially tough because it involves getting Lucy to talk about what she did, even though she’s doing it at a sort of visceral level, and then spell the words she needs to use — which is the other type of homework she has. I got her to say out loud what she was thinking, and then I had her say it back to me and then helped her spell the words.

So we made it through the worksheet, but there are a lot of questions in my mind about the pedagogical design of this stuff. And how in the world does this sort of thing work in a household where the parents don’t have the time, patience, interest, fluency, or comfort level in mathematics to sit down and work all this out with the kid?

Filed under Early education, Education, Math

## Small proof of concept for inverted math classrooms

Image via Wikipedia

The very last topic in the linear algebra class this semester (just concluded) unexpectedly gave me a chance to test-drive the inverted classroom model in a mathematics course, with pretty interesting results. The topic was least squares solutions and applications to linear models. I like to introduce this topic without lecture, since it’s really just an application of what they’ve learned about inner products and orthogonality. Two days are set aside for this topic. In the first day, I gave this group activity:

The intent was to get this activity done in about 35 minutes and then talk about the normal equations — a much faster way of finding the least-squares solution than what this activity entails — afterwards. Then I meant to spend a second day on practice.

But, as they say, life got in the way — and we ended up spending the entire time just getting through the first 3/4 of the activity. We spent most of the second day debriefing the math in the activity, reducing my lecture on the normal equations to a 10-minute time slot… which I then screwed up hugely by making an intractable math error which I didn’t catch until time was up. To make things worse, this was the last day I had budgeted for covering material. So this was going to be on the final, but I had no time in which to teach it right.

Solution: Make a video containing the correct form of the lecture I was going to give, and a couple minutes shorter to boot:

…make another video with a more intricate example:

…and then give students some exercises to do and open the email/Skype/IM/office hours line for questions.

By the night before the final exam, the videos had about 15 hits each, some of which were from IP addresses my students would have. So they were being watched. On the final, I gave a mid-level exercise from the section on linear models in their textbook — an even-numbered one, so it was fairly unlikely they worked it.

The results? Out of 14 students taking the exam:

• Two students had basically no progress on the problem — left it blank or just wrote down the problem.
• Five students did NOT use the normal equations approach from the videos but the projection-oriented approach from the group work. Of those five, only two used the method correctly.
• Seven students DID use the normal equations approach from the videos. Of those seven, six used the method correctly.

I’d like to conclude that the videos were more effective in conveying the subject matter than the group work was, and took less time. But of course there are confounding variables. The videos covered material that is less complicated than the group work covered. Perhaps only the most motivated students actually watched the videos, and their motivation rather than the video is what helped them to learn. But still, I think the inverted approach here had some kind of effect. Certainly the students who watched the videos were mostly (6 out of 7) able to pull off the method on the final, whereas less than half the students who relied upon what we did in class could do so with that other method.

Ideally, I’d put these videos out there, assign them along with some basic computation exercises, and then come in and do a group work assignment similar to what I used. With these kinds of results, I’m a little more emboldened to do so with some classes this fall. But more on that later.

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Filed under Inverted classroom, Linear algebra, Math, Teaching, Technology